## _wp08123

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---

### Introduction: objectives and headline findings
- Objective: evaluate degree of market integration among NAFTA members by analyzing relative price differentials across countries and sectors.
- Key empirical questions and findings:
  - Is the degree of Mexico–U.S. integration similar to that of Canada–U.S. integration?
    - Finding: transaction costs are larger for the Mexico–U.S. country pair than for the Canada–U.S. pair.
  - Have markets become more integrated, with reduced transaction costs, after the introduction of NAFTA?
    - Finding: NAFTA significantly reduced price differentials between the U.S. and Mexico, though this reduction was not uniform across sectors.
  - What determinants explain transaction costs?
    - Findings: sharing a common border and having lower exchange rate volatility reduce transaction costs; industry- or good-specific characteristics account for a large part of transaction costs.

### Theoretical underpinnings: LOOP, PPP, and transaction-cost bands
- Law of One Price (LOOP): identical goods should sell for the same price across countries when prices are expressed in a common currency; at the aggregate level this is purchasing power parity (PPP).
- Empirical regularity: relative prices often fail to fully converge and are persistent; relative price differentials are significant and highly correlated with exchange rate movements.
- Mechanism for nonconvergence: transaction costs (transport costs, explicit or implicit trade barriers) create a threshold band:
  - Inside the band: marginal costs of arbitrage exceed marginal benefits → zone of no-trade → sectoral real exchange rate (SRER) may follow a unit root (random walk) and prices need not equalize.
  - Outside the band: arbitrage profitable → SRER can become mean-reverting.
- Prior literature: nonlinearities in SRERs documented for developed markets; transaction costs vary across countries and sectors.

### Estimation methodology: SETAR model and interpretation
- Model class: Self-Exciting Threshold Autoregressive (SETAR) model, implemented as a TAR(1, 2, d) with delay d.
- Conceptual model elements:
  - Deviations from LOOP for sector j in country i at time t: x_{ijt} = s_{it} + p_{ijt} − p_{jt}.
  - Demeaned relative price q_{ijt} estimated as OLS residuals (q is the demeaned component of x).
  - Threshold parameter: κ_{ij}; threshold variable: lagged q with delay d.
  - Restriction: α restricted to unity so that within the threshold band the process is assumed to follow a random walk.
  - Outer regime characterized by autoregressive parameters ρ, and mean reversion occurs if 1ρ<.
- Interpretation:
  - Estimated threshold band size is a measure of transaction costs.
  - Presence of a unit root inside the band and mean reversion outside is consistent with arbitrage costs preventing equalization until deviations exceed transaction costs.

### Testing procedures and implementation details
- Three-step empirical approach:
  1. Test the validity of threshold model versus a null hypothesis of a random walk (unit root) to detect any degree of price convergence.
  2. Where convergence is present, test whether adjustment is nonlinear/asymmetric consistent with arbitrage (i.e., nonlinear model vs linear).
  3. For cases with nonlinear convergence pre- and post-NAFTA, test equality of threshold band size across periods using bootstrap inference.
- Estimation specifics:
  - Threshold and autoregressive parameters estimated simultaneously via least squares over a grid of κ.
  - Grid-search range: include observations between the 15th and 85th percentile of the threshold variable.
  - Delay parameter d considered over values from 1 to 4 months.
  - Choose κ and d that minimize the sum of squared residuals.
- Inferential procedure for likelihood-ratio statistic:
  - Likelihood ratio statistic: FT = T(σ_i^2/σ_l^2 − 1) where T is sample size, σ_i^2 and σ_l^2 are restricted and unrestricted residual variances (1/T times sum of squared residuals).
  - Bootstrap approximation steps:
    - Step 1: Estimate model under null.
    - Step 2: Generate T+100 observations from a data generating process calibrated to restricted estimates.
    - Step 3: Discard first 100 artificial observations; use remaining T to estimate restricted and unrestricted models.
    - Step 4: Compute bootstrap likelihood ratio statistic FT*.
    - Step 5: Repeat steps 2–5 1,000 times to construct empirical distribution.

### Data and empirical scope
- Data: disaggregated monthly consumer price indices (CPI) for 18 sectors, from January 1980 to December 2006, for Mexico, the United States, and Canada.
- Model: SETAR (1, 2, d) estimated simultaneously over k and d via grid search; tests include unit root in outer regime, linearity (Hansen test), and equality of thresholds pre- and post-NAFTA.
- Data sources: Bank of Mexico, U.S. Bureau of Labor Statistics, Statistics Canada; monthly nominal exchange rates are period averages from International Financial Statistics (IFS) of the IMF.
- Empirical innovation: application of recent testing techniques to SETAR models in the context of an emerging market (Mexico), and explicit testing of SETAR against a nonstationary (unit root) null.

### III. Estimation results — key findings (A. Testing for nonlinear price convergence)
- First-step unit root test determines whether SRER is nonstationary (unit root inside threshold band), implying disconnection of prices and failure of the LOOP.
- Exact outcomes:
  - "Half of the SRERs in the pre-NAFTA period followed a unit root process and only four of them in the post-NAFTA period."
- Model performance:
  - "In all cases the SETAR model clearly outperforms the linear one," confirming thresholds and allowing estimation of transaction costs.

### III. Estimation results — Estimated transaction costs (B)
- Sectoral pattern:
  - Highly homogenous sectors (for example, fish products and fruits) show low threshold bands.
- Mexico–U.S. SRERs:
  - Pre-NAFTA range of transaction costs across sectors: 7–32 percent.
  - Post-NAFTA range of transaction costs across sectors: 2–20 percent.
  - Several cases move from "infinite" (unit root) to measurable transaction costs post-NAFTA.
- Cross-country comparisons:
  - "Overall, average transaction costs among NAFTA members are 34 percent higher between the U.S. and Mexico than between the U.S. and Canada."
  - Comparison benchmark: threshold bands among NAFTA members are on average slightly lower than between the United States and European countries; when the United Kingdom is used as reference, estimated country average transaction costs range from 7 percent to 17 percent.

### III. Estimation results — Robustness checks (C)
- Detrending:
  - Demeaned series is preferred; results are robust to using detrended sectoral real exchange rates.
- Tequila crisis checks:
  - Robustness tests include (i) allowing different mean over 1994:12 to 1995:12 and (ii) restricting estimation to 1996–2006.
  - Results: "Overall, they are broadly consistent with the ones we discuss here, which reflects that Tequila crisis does not significantly affect our findings."
  - With allowance for different mean during Tequila crisis, cross-country average difference: "average transaction costs among NAFTA members are 27 percent higher between the U.S. and Mexico than between the U.S. and Canada."

### III. Estimation results — Half-lives of relative price adjustments (D)
- Definition and computation:
  - Half-life: time for 50 percent of a shock to die out.
  - Computed accounting for regime-switching SETAR model; half-life is infinite within threshold band and equal to ln(0.5)/ln(ρ) outside the band.
  - Half-lives for 10 percent, 20 percent, 30 percent, 40 percent, and 50 percent shocks computed by stochastic simulation using generalized impulse response functions (Koop and others, 1996).
- Key quantitative results:
  - Mexico–U.S. pair, 10 percent shock:
    - Average half-life pre-NAFTA: 20 months.
    - Average half-life post-NAFTA: 11 months.
  - Cross-pair post-NAFTA comparison for 10 percent shock:
    - Mexico–U.S. average half-life: 11 months.
    - Canada–U.S. average half-life: 12 months.
- Distributional observations:
  - Post-NAFTA: less variation across shock sizes than pre-NAFTA; relative prices adjust more quickly irrespective of shock size.
  - In post-NAFTA period, almost 60 percent of SRERs adjust (by half) to a 10 percent shock within 6 to 9 months.
  - In pre-NAFTA period, most (70 percent) SRERs take more than a year to adjust, and 55 percent take more than 18 months.
  - Half-lives vary substantially across sectors: homogenous goods (food products) adjust faster; high-end products (furniture, photographic equipment) adjust more slowly.

### Box 1 — Real Exchange Rate Thresholds at the Aggregate CPI Level
- Estimation approach and scope:
  - Convergence thresholds estimated at the national CPI (aggregate) index level using SETAR (1, 2, d).
  - p-value Ha, p-value Hb and p-value Hc represent marginal significance levels of (i) unit root in the outer regime, (ii) linearity, and (iii) equality of thresholds pre- and post-NAFTA.
- Main aggregate-level findings:
  - For all three country pairs, evidence of nonlinear convergence of aggregate price levels for both pre- and post-NAFTA periods.
  - Size of the thresholds is significantly smaller in the post-NAFTA period.
  - Thresholds are smaller for the U.S.–Canada country pair than for the U.S.–Mexico pair.
  - After NAFTA, the estimated thresholds are reduced:
    - U.S.–Canada: from 13 percent to 10 percent.
    - U.S.–Mexico: from 18 percent to 14 percent.
  - The finding of thresholds at the aggregate level suggests sectoral limitations to price convergence can be macroeconomically significant.
- Determinants of thresholds (gravity-style regression; sample N: 94):
  - Reported coefficient estimates (columns (1) and (2)):
    - Distance: -0.041 * (column (1)), -0.040 * (column (2))
    - Dummy post NAFTA: -0.109 ** (column (1)), -0.116 ** (column (2))
    - Exchange Rate Volatility: 4.020 *** (column (1)), 3.991 *** (column (2))
    - Firms: -0.001- (column (1))  [not significant]
    - Tradability: -0.042- (column (1))  [not significant]
    - R2: 0.33 (both columns)
    - N: 94 (both columns)
  - Significance markers: *** indicates a 1 percent degree of confidence, ** indicates a 5 percent degree of confidence, * indicates a 10 percent degree of confidence.
- Interpretation:
  - Three variables appear significant: the post-NAFTA dummy (negative), the border effect (common-border dummy), and exchange rate volatility (positive).
  - Post-NAFTA dummy confirms that introduction of NAFTA reduced transaction costs.
  - Exchange rate volatility is positively associated with higher thresholds.
  - Sectoral proxies for competition (number of firms) and tradability were not statistically significant in this sample.
- Summary statistics and dynamics:
  - Estimated price thresholds across sectors for Mexico–U.S. and Canada–U.S. range from 2 percent to 32 percent.
  - Average transaction costs among NAFTA members are 34 percent higher between the U.S. and Mexico than between the U.S. and Canada.
  - Average half-life of Mexico–U.S. sectoral real exchange rate deviations:
    - Pre-NAFTA: 20 months.
    - Post-NAFTA: 11 months.
  - Exchange-rate fluctuation context for Mexico: maximum and minimum monthly averages of the peso against the U.S. dollar have differed by only about 10 percent in the last several years—smaller than the transaction cost bands found for many sectors—implying limited pass-through of such exchange rate changes to domestic prices for many traded goods.

### Policy implications and conclusions
- Emerging markets (Mexico) still face higher transaction costs than developed counterparts.
- Trade liberalization (NAFTA) helped lower transaction costs and speed adjustment of relative prices.
- Possible avenues to reduce remaining transaction costs include developing logistics, transportation, and internal distribution mechanisms, and enhancing competition among domestic firms and reducing remaining barriers to external trade.
- Further analysis is encouraged to understand why Mexico–U.S. transaction costs continue to exceed Canada–U.S. costs for many goods and whether policy actions can reduce these costs.

*Source: IMF working paper chapter — 1. Extent of Price Convergence between Mexico–U.S. and Canada–U.S. (from the supplied content unit).*

### 1.   Extent of Price Convergence between Mexico–U.S. and Canada–U.S. ............................10

### 1.   Extent of Price Convergence between Mexico–U.S. and Canada–U.S. ............................10

### Introduction: objectives and headline findings
- Objective: evaluate degree of market integration among NAFTA members by analyzing relative price differentials across countries and sectors.
- Key empirical questions and findings:
  - Is the degree of Mexico–U.S. integration similar to that of Canada–U.S. integration?
    - Finding: transaction costs are larger for the Mexico–U.S. country pair than for the Canada–U.S. pair.
  - Have markets become more integrated, with reduced transaction costs, after the introduction of NAFTA?
    - Finding: NAFTA significantly reduced price differentials between the U.S. and Mexico, though this reduction was not uniform across sectors.
  - What determinants explain transaction costs?
    - Findings: sharing a common border and having lower exchange rate volatility reduce transaction costs; industry- or good-specific characteristics account for a large part of transaction costs.

### Theoretical underpinnings: LOOP, PPP, and transaction-cost bands
- Law of One Price (LOOP): identical goods should sell for the same price across countries when prices are expressed in a common currency; at the aggregate level this is purchasing power parity (PPP).
- Empirical regularity: relative prices often fail to fully converge and are persistent; relative price differentials are significant and highly correlated with exchange rate movements.
- Mechanism for nonconvergence: transaction costs (transport costs, explicit or implicit trade barriers) create a threshold band:
  - Inside the band: marginal costs of arbitrage exceed marginal benefits → zone of no-trade → sectoral real exchange rate (SRER) may follow a unit root (random walk) and prices need not equalize.
  - Outside the band: arbitrage profitable → SRER can become mean-reverting.
- Prior literature overview (as presented): nonlinearities in SRERs documented for developed markets; transaction costs vary across countries and sectors.

### Estimation methodology: SETAR model and interpretation
- Model class: Self-Exciting Threshold Autoregressive (SETAR) model, implemented as a TAR(1, 2, d) with delay d.
- Conceptual model elements:
  - Deviations from LOOP for sector j in country i at time t: x_{ijt} = s_{it} + p_{ijt} − p_{jt}.
  - Demeaned relative price q_{ijt} estimated as OLS residuals (q is the demeaned component of x).
  - Threshold parameter: κ_{ij}; threshold variable: lagged q with delay d.
  - Restriction: α restricted to unity so that within the threshold band the process is assumed to follow a random walk (persistent deviations).
  - Outer regime characterized by autoregressive parameters ρ, and mean reversion occurs if 1ρ<.
- Model simplification under symmetric adjustments outside the band leads to the specified SETAR difference equation (Equation (5) in source).
- Interpretation:
  - Estimated threshold band size is a measure of transaction costs.
  - Presence of a unit root inside the band and mean reversion outside is consistent with arbitrage costs preventing equalization until deviations exceed transaction costs.

### Testing procedures and implementation details
- Strategy: three-step empirical approach
  1. Test the validity of threshold model versus a null hypothesis of a random walk (unit root) to detect any degree of price convergence.
  2. Where convergence is present, test whether adjustment is nonlinear/asymmetric consistent with arbitrage (i.e., nonlinear model vs linear).
  3. For cases with nonlinear convergence pre- and post-NAFTA, test equality of threshold band size across periods using bootstrap inference because conventional tests have asymptotic nonstandard distributions.
- Estimation specifics:
  - Threshold and autoregressive parameters estimated simultaneously via least squares over a grid of κ.
  - Grid-search range: include observations between the 15th and 85th percentile of the threshold variable.
  - Delay parameter d considered over values from 1 to 4 months.
  - Choose κ and d that minimize the sum of squared residuals.
- Inferential procedure for likelihood-ratio statistic:
  - Likelihood ratio statistic: FT = T(σ_i^2/σ_l^2 − 1) where T is sample size, σ_i^2 and σ_l^2 are restricted and unrestricted residual variances (1/T times sum of squared residuals).
  - Bootstrap approximation of asymptotic distribution: 
    - Step 1: Estimate model under null.
    - Step 2: Generate T+100 observations from a data generating process calibrated to restricted estimates.
    - Step 3: Discard first 100 artificial observations; use remaining T to estimate restricted and unrestricted models.
    - Step 4: Compute bootstrap likelihood ratio statistic FT*.
    - Step 5: Repeat steps 2–5 1,000 times to construct empirical distribution.

### Empirical innovation and scope
- Innovation: application of recently developed testing techniques to SETAR models in the context of an emerging market (Mexico), and explicit testing of SETAR against a nonstationary (unit root) null.
- Scope: sectoral real exchange rates across NAFTA members, with attention to pre- and post-NAFTA period comparisons and sectoral heterogeneity in transaction costs.

*Source: IMF working paper chapter — 1.   Extent of Price Convergence between Mexico–U.S. and Canada–U.S.*

### 6.   The asymptotic approximation to the bootstrap p-value of the test is calculated by the

### _wp08123 - 6.   The asymptotic approximation to the bootstrap p-value of the test is calculated by the

### III. Estimation results — data and approach
- Data: disaggregated monthly consumer price indices (CPI) for 18 sectors, from January 1980 to December 2006, for Mexico, the United States, and Canada.
- Model: SETAR (1, 2, d) estimated simultaneously over k and d via grid search; tests include unit root in outer regime, linearity (Hansen test), and equality of thresholds pre- and post-NAFTA.
- Data sources: Bank of Mexico, U.S. Bureau of Labor Statistics, Statistics Canada; monthly nominal exchange rates are period averages from International Financial Statistics (IFS) of the IMF.

### A. Testing for nonlinear price convergence — key findings
- First-step unit root test determines whether SRER is nonstationary (unit root inside threshold band), implying disconnection of prices and failure of the LOOP.
- Exact outcomes reported in text:
  - "Half of the SRERs in the pre-NAFTA period followed a unit root process and only four of them in the post-NAFTA period."
- SETAR model performance:
  - "In all cases the SETAR model clearly outperforms the linear one," confirming thresholds and allowing estimation of transaction costs.

### B. Estimated transaction costs — sectoral and cross-country results
- Sectoral pattern:
  - Highly homogenous sectors (for example, fish products and fruits) show low threshold bands.
- Mexico–U.S. SRERs:
  - Pre-NAFTA range of transaction costs across sectors: 7–32 percent.
  - Post-NAFTA range of transaction costs across sectors: 2–20 percent.
  - Several cases move from "infinite" (unit root) to measurable transaction costs post-NAFTA.
- Cross-country comparisons:
  - "Overall, average transaction costs among NAFTA members are 34 percent higher between the U.S. and Mexico than between the U.S. and Canada."
  - Comparison to other work: threshold bands among NAFTA members are on average slightly lower than between the United States and European countries; when the United Kingdom is used as reference, estimated country average transaction costs range from 7 percent to 17 percent (noted as a benchmark for comparison).

### C. Robustness checks — detrending and Tequila crisis
- Detrending:
  - Demeaned series is preferred; results are robust to using detrended sectoral real exchange rates. Stability suggests trend component may not be crucial.
- Tequila crisis (structural break) checks:
  - Robustness tests include (i) allowing different mean over 1994:12 to 1995:12 and (ii) restricting estimation to 1996–2006.
  - Results: "Overall, they are broadly consistent with the ones we discuss here, which reflects that Tequila crisis does not significantly affect our findings."
  - With allowance for different mean during Tequila crisis, cross-country average difference: "average transaction costs among NAFTA members are 27 percent higher between the U.S. and Mexico than between the U.S. and Canada."

### D. Half-lives of relative price adjustments — methodology and results
- Definition and computation:
  - Half-life: time for 50 percent of a shock to die out.
  - Computed accounting for regime-switching SETAR model; half-life is infinite within threshold band and equal to ln(0.5)/ln(ρ) outside the band.
  - Half-lives for 10 percent, 20 percent, 30 percent, 40 percent, and 50 percent shocks computed by stochastic simulation using generalized impulse response functions (Koop and others, 1996).
- Key quantitative results:
  - Mexico–U.S. pair, 10 percent shock:
    - Average half-life pre-NAFTA: 20 months.
    - Average half-life post-NAFTA: 11 months.
  - Cross-pair post-NAFTA comparison for 10 percent shock:
    - Mexico–U.S. average half-life: 11 months.
    - Canada–U.S. average half-life: 12 months.
- Distributional observations:
  - Post-NAFTA: less variation across shock sizes than pre-NAFTA; relative prices adjust more quickly irrespective of shock size.
  - In post-NAFTA period, almost 60 percent of SRERs adjust (by half) to a 10 percent shock within 6 to 9 months.
  - In pre-NAFTA period, most (70 percent) SRERs take more than a year to adjust, and 55 percent take more than 18 months.
  - Half-lives vary substantially across sectors: homogenous goods (food products) adjust faster; high-end products (furniture, photographic equipment) adjust more slowly.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08123.pdf*

### Box 1. Real Exchange Rate Thresholds at the Aggregate CPI Level

### Box 1. Real Exchange Rate Thresholds at the Aggregate CPI Level

### Estimation approach and scope
- Convergence thresholds were estimated at the national CPI (aggregate) index level using a SETAR (1, 2, d) model as in equation (5).  
- Estimation of k, ρ and d was done simultaneously via a grid search over k and d.  
- p-value Ha, p-value Hb and p-value Hc represent, respectively, the marginal significance levels of (i) the null hypothesis of unit root in the outer regime, (ii) the null hypothesis of linearity, and (iii) the null hypothesis of equality of thresholds pre- and post- NAFTA.

### Main aggregate-level findings
- For all three country pairs, there is evidence of nonlinear convergence of aggregate price levels for both the pre- and post-NAFTA periods.  
- The size of the thresholds is significantly smaller in the post-NAFTA period.  
- Thresholds are smaller for the U.S.–Canada country pair than for the U.S.–Mexico pair.  
- After NAFTA, the estimated thresholds are reduced:
  - U.S.–Canada: from 13 percent to 10 percent.
  - U.S.–Mexico: from 18 percent to 14 percent.  
- The finding of thresholds at the aggregate level suggests that limitations to price convergence at the sectoral level can also be an issue of macroeconomic significance.

### SETAR estimation (aggregate CPI) — reported parameters
- The table reports estimated threshold (k) and outer regime root (ρ) alongside the p-values for unit-root, linearity, and equality-of-thresholds tests for pre- and post-NAFTA periods. (See Table 5 for full SETAR estimation results as presented in the source.)

### Determinants of thresholds (gravity-style regression results — Table 6)
- Regression specification: transaction costs (threshold κij) explained by a set of explanatory variables including a common-border dummy, exchange rate volatility, tradability, number of establishments, and a post-NAFTA dummy.  
- Sample: all three country pairs (including Mexico–Canada), pre- and post-NAFTA periods; 94 thresholds used as observations.  
- Reported coefficient estimates (columns (1) and (2) in Table 6) and associated significance markers:
  - Distance: -0.041 * (column (1)), -0.040 * (column (2))
  - Dummy post NAFTA: -0.109 ** (column (1)), -0.116 ** (column (2))
  - Exchange Rate Volatility: 4.020 *** (column (1)), 3.991 *** (column (2))
  - Firms: -0.001- (column (1))  [not significant]
  - Tradability: -0.042- (column (1))  [not significant]
  - R2: 0.33 (both columns)
  - N: 94 (both columns)
- Notes on significance markers from the table: *** indicates a 1 percent degree of confidence, ** indicates a 5 percent degree of confidence, * indicates a 10 percent degree of confidence.

### Interpretation of determinant results
- Three variables appear significant: the post-NAFTA dummy, the border effect (proxied in the paper by a common-border dummy), and exchange rate volatility.  
- The post-NAFTA dummy is strongly significant and negative, confirming that the introduction of NAFTA reduced transaction costs.  
- Exchange rate volatility is positively associated with higher thresholds (transaction costs).  
- Sectoral proxies for competition (number of firms) and tradability were not statistically significant in this sample; data quality and proxy limitations are cited as probable explanations.

### Summary statistics and dynamics highlighted in conclusions
- Estimated price thresholds across sectors for Mexico–U.S. and Canada–U.S. range from 2 percent to 32 percent.  
- Average transaction costs among NAFTA members are 34 percent higher between the U.S. and Mexico than between the U.S. and Canada.  
- Average half-life of Mexico–U.S. sectoral real exchange rate deviations:
  - Pre-NAFTA: 20 months.
  - Post-NAFTA: 11 months.
- Exchange-rate fluctuation context for Mexico: maximum and minimum monthly averages of the peso against the U.S. dollar have differed by only about 10 percent in the last several years—smaller than the transaction cost bands found for many sectors—implying limited pass-through of such exchange rate changes to domestic prices for many traded goods.

### Policy implications and conclusions
- Results support that emerging markets (Mexico) still face higher transaction costs than developed counterparts, and that trade liberalization (NAFTA) helped lower transaction costs and speed adjustment of relative prices.  
- Possible avenues to reduce remaining transaction costs include developing logistics, transportation, and internal distribution mechanisms, and enhancing competition among domestic firms and reducing remaining barriers to external trade.  
- Further analysis is encouraged to understand why Mexico–U.S. transaction costs continue to exceed Canada–U.S. costs for many goods and whether policy actions can reduce these costs.

*Source: Box 1. Real Exchange Rate Thresholds at the Aggregate CPI Level (from the supplied IMF content unit).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08123.pdf_